This is a summary of the governing equations I've learned by far.
The ideas of mechanics and mathematical physics may be extended into social systems, such as economics (input-output model) and urban science.
To capture the dynamics of a system, constraints are put forward to form a closed set of equations:
A continuity equation represents a conservation law.
Denote \( j^\mu \) as a conserved current, the general continuity equation reads: \[ \partial_\mu j^\mu=0 \]
Table 1: Pairs of conserved currents and corresponding continuity equations
| Conserved current | Continuity equations |
|---|---|
| Mass flux | \[ {\partial \rho \over \partial t} + \nabla \cdot (\rho \mathbf{u}) = 0 \] |
| Energy flux | \[ \nabla \cdot \mathbf{q} + \frac{ \partial u}{\partial t} = 0 \] |
| Electric current | \[ \nabla \cdot \mathbf{J} = - {\partial \rho \over \partial t} \] |
| Probability current | \[ \nabla \cdot \mathbf{j} + \frac{\partial \lvert \Psi \rvert^2}{\partial t} = 0 \] |
| Current of particles in phase space | \[ \frac{\partial\rho}{\partial t}+\nabla \mathbf{J}=0 \], where \( J = (\rho\dot{q}^i,\rho\dot{p}_i) \) and \( \rho = \rho(q^i, p_i) \) |
Justification of prevalence of conservation laws:
By introducing force, we could write time derivative of (conserved) quantities as equations of motion.
Table 2: Equations of Motion and Equilibrium in Mechanics
| Field of study | Equations of motion/equilibrium |
|---|---|
| Classic mechanics | \[\frac{d p }{d t} = F\] \[\frac{\mathrm{d}\mathbf{L}}{\mathrm{d}t} = \boldsymbol{\tau}\] \[\text{d} T = \text{d} W_{ext} + \text{d} W_{int}\] |
| Elasticity | \[ \boldsymbol{\nabla}\cdot\boldsymbol{\sigma} + \mathbf{F} = \rho\ddot{\mathbf{u}} \] |
| Fluid mechanics | \[ \rho\frac{D\mathbf{v}}{D t} = \nabla\cdot\mathbb{P} + \rho\mathbf{f} \] |
Some dismiss the concept of force.
Because force is the time derivative of momentum, the concept of force is redundant and subordinate to the conservation of momentum, and is not used in fundamental theories.
While arguments also go against the perception of quantity conservation.
The discovery of the Second Law of Thermodynamics by Carnot in the 19th century showed that every physical quantity is not conserved over time. ... Hence, a "steady-state" worldview based solely on Newton's laws and the conservation laws does not take entropy into account.
Reynolds' transport theorem in continuum mechanics:
\[ \frac{\mathrm{d}}{\mathrm{d}t}\int_{\tau}{\varphi}~{\text{d}\tau} = \frac{\partial}{\partial t} \int_{\text{CV}}{ \varphi ~{\text{d}\tau} } + \int_{\text{CS}} \varphi ({\mathbf{v}}^{r}\cdot {\mathbf{n}}) ~{\text{d}A} \]
The theorem is a generalization of the Leibniz integral rule (a.k.a. differentiation under the integral sign).
Consistency and compatibility conditions in elasticity:
Linear relations between polarizations and electromagnetic fields:
Polarization P is the electric dipole moment per unit volume.
Magnetization M is the magnetic dipole moment per unit volume.
The relations between electromagnetic fields and polarization or magnetization depend on how the dipoles respond to the applied fields.
When the applied fields are weak, to first order approximation, the dipole moments will have a linear relation to the field.
Here \( \varepsilon_0 \) is the electric constant, aka vacuum permittivity; \( \chi_e \) is electric susceptibility; \( \chi_m \) is magnetic susceptibility; \(\mathbf{H}\) is the auxiliary magnetic field, defined by \(\mathbf{B}=\mu_0(\mathbf{H + M})\).
There are several laws which describe the transport of matter, or properties of it, in an almost identical way. In every case, in words they read:
Flux (density) is proportional to a gradient, the constant of proportionality is the characteristic of the material.
Table 3: Linear Constitutive Relations of Transport Phenomena
| Name of Law | Mathematical Expression |
|---|---|
| Fick's law of diffusion | \[ \mathbf{J}=- D\nabla \phi \] |
| Newton's law of viscosity | \[ \tau=\mu \frac{\partial u}{\partial y} \] |
| Fourier's law of thermal conduction | \[ \mathbf{q} = - k {\nabla} T \] |
| Ohm's law of electric conduction | \[ \mathbf{J} = \sigma \mathbf{E} \] |
| Darcy's law for porous flow | \[ q = -\frac{k}{\mu} \nabla P \] |
Fluid mechanics (Newtonian fluid): \[ \tau_{ij} = 2\mu (s_{ij} - \tfrac{1}{3} s_{kk} \delta_{ij}) \]
Linear Elasticity (tensor and isotropic form):
\[ \boldsymbol{\sigma} = \mathsf{C}:\boldsymbol{\varepsilon} \] \[ \sigma_{ij} = \lambda \varepsilon_{kk} \delta_{ij} + 2\mu \varepsilon_{ij} \]
Plasticity: Several nonlinear constitutive relations are presented, either in functional form or differential form. Omitted here for the sake of time.
Dynamic (shear) viscosity \( \mu \) is equivalent to shear modulus \( \mu \); in the sense of linear isotropic materials both of which are Lamé's second parameters. The elimination of second viscosity (bulk viscosity) is called Stokes assumption.
Eventually, we can close a system by collection all the above into a set of governing equations.
Navier–Stokes equations for fluid mechanics:
\[ \rho \left({\frac {\partial {\mathbf {v}}}{\partial t}}+{\mathbf {v}}\cdot \nabla {\mathbf {v}}\right)=-\nabla p+\nabla \cdot {\boldsymbol {{\mathsf {T}}}}+{\mathbf {f}} \]
Schrödinger equation for quantum physics:
\[ i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},t) = \left [ \frac{-\hbar^2}{2m}\nabla^2 + V(\mathbf{r},t)\right ] \Psi(\mathbf{r},t) \]
Maxwell's equations for electrodynamics: